Arsenal are in no rush to sanction Walcott’s exit – they want to asses the condition of Olivier Giroud after a hamstring injury and fellow striker Danny Welbeck following a series of problems.

If Giroud and Welbeck prove their fitness and Arsenal suffer no other major setbacks, it is likely Walcott will leave 12 years after joining from Southampton.

Returning to his first professional team has always appealed to Walcott, though Everton’s ambition to challenge the established top six of Arsenal, Chelsea, Liverpool, Manchester City, Manchester United and Tottenham is also understood to be an attraction.

Walcott was selected by Toffees boss Sam Allardyce during his short spell as England manager and is thought to be enthusiastic about the idea of playing under him again.

Everton are also understood to be more likely than Southampton to pay the transfer fee being sought by the Gunners – reportedly about £20m – and Walcott’s £110,000-per-week salary.

Another factor in any decision could be Everton’s position in the Premier League – they are in ninth place, while Southampton are only one place above the relegation zone.

Walcott arrived at Arsenal in January 2006 – for £5m rising to about £12.5m – and has scored 108 goals in 396 appearances, with eight goals in 47 games for England.

Earlier on Thursday, Saints boss Mauricio Pellegrino said: “We know Walcott is a very good player but now it is not my concern to talk about players not in the club.

“We have to be optimistic. We have a couple of targets and we are working hard in this way. The priority now is to try to help, especially in the final third.

“But the market is really difficult. It’s about negotiations – we can control our side but not the other side.”